In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the bounded compact approximation property, then L∞/X has the bounded compact approximation property. We also show that X∗ has the λ-BCAP with conjugate operators if and only if the pair (X,Y ) has the λ-BCAP for each...

Let μ be a non-negative Radon measure on R which only satisfies the polynomial growth condition. Let Y be a Banach space and H(μ) the Hardy space of Tolsa. In this paper, the authors prove that a linear operator T is bounded from H(μ) to Y if and only if T maps all (p, γ)-atomic blocks into uniformly bounded elements of Y; moreover, the authors prove that for a sublinear operator T bounded from...

Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ...

Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ...

The main purpose of this paper is to derive a new (p, q)-atomic decomposition on the multi-parameter Hardy space H(X1 ×X2) for 0 < p0 < p ≤ 1 for some p0 and all 1 < q < ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L(X1 × X2) (for 1 < q < ∞) and Hardy space H(X1 × X2) (for 0 < p ≤ 1). As an application,...

We prove that any weak space-time L vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of R satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that t − a.e. weak L inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they...

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617-637]. Therein, the key assumption is that the conforming first-order finite element space V c h annulates the linear and bounded residual l written V c h ⊆ ker l. That excludes ...

We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator (L,D(L)) in the space of all uniformly continuous and bounded functions f : H → R, where H is a separable Hilbert space. We consider a perturbation of the form N0φ = Lφ + 〈Dφ,F 〉 where F : H → H is bounded and Fréchet differentiable with uniformly continuous and bounded differential. Hence, we prove that N...

For an operator generating a group on $$L^p$$ spaces transference results give bounds the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific generators which are abstraction of first order differential operators and prove similar estimates assuming only that is bounded $$L^2$$ rather than . We R-bounded Hörmander result by abstract Sobol...

We extend Lutz’s resource-bounded measure to probabilistic classes, and obtain notions of resource-bounded measure on probabilistic complexity classes such as BPE and BPEXP. Unlike former attempts, our resource bounded measure notions satisfy all three basic measure properties, that is every singleton {L} has measure zero, the whole space has measure one, and “enumerable infinite unions” of mea...